The Electronic Journal of Combinatorics

v9i1r26 — Comment by authors, Sep 6, 2002 (edited Feb 13, 2012).

After our paper was published, we learned, through James Fill, that the same Markov chain was studied more than two decades ago by Kingman in the context of the "common ancestor problem." The upper bound in Theorem 5 of Kingman [7] is already better than ours, and in the intervening years the result has been generalized and extended. See, for example, Theorem 3 of Donnelly[1], and Möhle[9],[10]. (We gratefully acknowledge Simon Tavaré's help in locating these references.)  There is an enormous literature that can be traced back to Kingman's work . We do not attempt a review here, but simply acknowledge our lack of priority.

  1. P. Donnelly, Weak convergence to a Markov chain with an entrance boundary: ancestral processes in population genetics, The Annals of Probability 19 No.3, (1991) 1102-1117.
  2. P. Donnelly and S. Tavaré, Coalescents and genealogical structure under neutrality, Annual Review of Genetics 29 (1995) 401-425.
  3. R.C. Griffiths, Exact sampling distributions from the infinite neutral alleles model, Advances in Applied Probability 11 (1979) 326-354.
  4. R.C. Griffiths, Lines of descent in the diffusion approximation of neutral Wright-Fisher models, Theoretical Population Biology 17 (1980) 37-50.
  5. J.F.C. Kingman, The Coalescent, Stochastic Proc. Appl.  13 (1982) 235-248.
  6. J.F.C. Kingman, On the genealogy of large populations. Essays in statistical science. J. Appl. Probab. 19A (1982) 27-43.
  7. J.F.C. Kingman, Exchangeability and the evolution of large populations, in Exchangeability in probability and statistics, pp. 97-12, North-Holland, Amsterdam-New York, 1982.
  8. J.F.C. Kingman, Mathematics of genetic diversity. CBMS-NSF Regional Conference Series in Applied Mathematics 34. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1980.
  9. M. Möhle, The time back to the most recent common ancestor in exchangeable population models, Adv. in Appl. Probab. 36(1):78-97, 2004.
  10. M. Möhle, Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models, Adv. Appl. Prob. 32 (2000) 983-993.
  11. S. Tavaré, Line-of-descent and genealogical processes and their applications in population genetics models, Theoretical Population Biology 26 (1984) 119-164.
  12. S. Tavaré , Ancestral inference from DNA Sequence data, Statistical Science 4 No.3 (1994) 307-319.
  13. G.A. Watterson, On the number of segregating sites in genetic models without recombination, Theoretical Population Biology 7 (1975) 256-276.